What is Small angle: Definition and 32 Discussions

The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:








sin

θ




θ




cos

θ




1




θ

2


2



1




tan

θ




θ






{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\cos \theta &\approx 1-{\frac {\theta ^{2}}{2}}\approx 1\\\tan \theta &\approx \theta \end{aligned}}}
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.
There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation,




cos

θ



{\displaystyle \textstyle \cos \theta }
is approximated as either



1


{\displaystyle 1}
or as



1




θ

2


2




{\textstyle 1-{\frac {\theta ^{2}}{2}}}
.

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  1. S

    De Broglie Wavelength problem

    I tried using lamba = h/p as follows: (6.626 * 10^-34 J *s) / (8 m.s * 0.6 kg) = 1.38041667*10^−34 and then using the small angle approximation sin(alpha) = lamba/d as follows: (1.38041667*10^−34)/(0.6m) = 2.30069444 * 10^−34 then converting to radians with the following: (2.30069444 *...
  2. Beth N

    Diffraction Grating Vs Double-Slit: Small angle Approx.

    Homework Statement In double slit interference , the angle between the center bright fringe and the path length (aka the angle used to find the path length difference) can be approximate as a small angle. However, we cannot assume the angles of bright fringes due to diffraction gratings are...
  3. cookiemnstr510510

    Calculating the capacitance of a system of two plates tilted at a small angle

    Homework Statement A capacitor is made up of a pair of square plates, each of side length L, that are very nearly parallel (such that θ is very small) and separated by a distance D. The plates make an angle, θ, with one another. The capacitor may be thought of as being comprised of many...
  4. Jayalk97

    Quick question about linearization using the small angle method

    Hey guys, when you're linearizing a function that has a constant, what do you do to it? An example would be y = x^2 + 3, would you just linearize it using its derivative and get rid of the constant?
  5. F

    Small angle approximation

    Homework Statement I want to solve the motion equation ## m \frac {dv_z} {dt} = - μ \frac {∂B_z} {∂z} ## with small angle approximation Homework Equations ## B_z(z) = B_0 -bCos(\frac {zπ} {2L}) ## is the magnetic field in the z-direction The Attempt at a Solution Started by derive the...
  6. Mr Davis 97

    I Small angle expansions for sin, cos, and tan

    From the Wikipedia article https://en.wikipedia.org/wiki/Small-angle_approximation, it says that they are "second-order approximations." What makes all three second order? Shouldn't sin and tan be first-order and cos be second-order?
  7. saadhusayn

    Pendulum oscillating in an accelerating car

    We have a car accelerating at a uniform rate ## a ## and a pendulum of length ## l ## hanging from the ceiling ,inclined at an angle ## \phi ## to the vertical . I need to find ##\omega## for small oscillations. From the Lagrangian and Euler-Lagrange equations, the equation of motion is given...
  8. FallenApple

    Small Angle Approximation to Hoop Oscillator

    Homework Statement A point particle of mass m slides without friction within a hoop of radius R and mass M. The hoop is free to roll without slipping along a horizontal surface. What is the frequency of small oscillations of the point mass, when it is close to the bottom of the hoop...
  9. H

    I Does small angle mean small angular velocity?

    Why is the term ##(\sin\theta\,\dot{\theta})^2## fourth order in the small ##\theta##, as claimed by the sentence below (5.108)? By small-angle approximation, ##(\sin\theta\,\dot{\theta})^2\approx\theta^2\,\dot{\theta}^2##. For this to be fourth order, it seems like we must have...
  10. D

    Time of oscillation of a pendulum

    Homework Statement A rigib poll of length 2L is made into a V shape so that each leg has length L. What is the period of oscillation for small angle. The angle between the legs is 120 degrees Homework Equations 3. The Attempt at a Solution [/B] I tried to calculate the period by imagining a...
  11. lulzury

    Small Angle Approximation in Single Slit Interference

    Homework Statement A monochromatic light source is used with a double slit to create an interference pattern on a screen that is 2.00 meters away. If the 2nd bright spot is observed 8.73 mm above the central maximum, can the small angle approximation be used? Show and/or explain your reasoning...
  12. A

    Explanation of small angle approximation?

    < Mentor Note -- The OP's question was in the thread title. They have been advised to please make more detailed and clear thread starters > thanks
  13. R

    Small Angle Approximation with a Stiff String

    Hi all, Could someone please help me understand a small but significant step in the derivation of the wave equation for a string with stiffness. I am trying to follow the notes here: http://courses.physics.illinois.edu/phys406/Lecture_Notes/Waves/PDF_FIles/Waves_2.pdf The statement...
  14. G

    Small angle approx. optics

    Homework Statement Had the same problem as this threadstarter: https://www.physicsforums.com/showthread.php?t=109059 Homework Equations The Attempt at a Solution I managed to find a ratio of tangents for the two angles. From there, it seems you're supposed to go "well tan(x)...
  15. A

    Small Angle Approximation

    Homework Statement Find, by comparison with exact trigonometry, the angle,  (provide a numerical value in degrees), above which the small angle approximation departs from the exact result by more than 1 percent. Homework Equations Approx.: d = s = rθ Exact: d = 2*r*Sin(θ/2) The...
  16. K

    Require help understanding small angle approximation

    Homework Statement I was recently given this question and very little explanation of the concept. I've struggled with this for a week and read absolutely everything I can find and I'm still not any closer to understanding it. Can anyone please point me in the right direction or explain how...
  17. V

    Small angle approximation for (dθ/dt)^2=0

    Hey guys, I am looking for a textbook that I can cite as a source for a project, for which I am doing the math on. I know that for a 22° approximation sinθ=θ and cosθ=1-\frac{θ^{2}}{2} but for a 5° approximation sinθ=θ but now cosθ=1 and that's all fine and dandy, but I am looking...
  18. A

    Relativistic collision in CM frame at small angle

    Homework Statement Find CM energy of a mu+ mu- collider, with each beam having an energy E of 500 GeV. The beams cross at a small angle of 250 mrad.Homework Equations E^2 - p^2c^2 = m^2c^4The Attempt at a Solution So I have a diagram for the lab frame which has the mu- coming in at a small...
  19. L

    Semi-circle with small angle harmonic oscillations

    Homework Statement Consider a uniform semicircular disk of radius R, which rolls without slipping on a horizontal surface. Recall that the kinetic energy of an object is the sum of the translational kinetic energy of the centre of mass (point C) and the rotational kinetic energy about the...
  20. genxium

    String wave and small angle approximation?

    Homework Statement I'm so confused about the derivation of the famous equation v=\sqrt{\frac{T}{u}}, I tried to derive it by myself but failed, then I turned to wikipedia but the derivation there really gave me a shock! http://en.wikipedia.org/wiki/Vibrating_string" [Broken] I have no...
  21. A

    Small Angle Question in relation to Astronomy

    First of all this is NOT for class! Secondly, this seems rather simple, so perhaps I am simply overlooking the obvious. Anyway, here goes: If some binary star is about 400 ly away, and a telescope gives a separation of 36" (0.01 degrees) for the binary, then can we use the small angle formula...
  22. P

    Double Slit Experiment without the small angle approximation

    Two very narrow slits are spaced 1.8 \mum apart and are placed 35.0 cm from the screen. What is the distance from the first and second dark lines of the interference pattern when the slits are illuminated with coherent light with \lambda =550 nm r1-r2=m\lambda dsin\theta=m\lambda...
  23. A

    Why is a small angle considered a vector?

    I have been told by my teacher that the angle of rotation, namely theta cannot be considered as a vector, which is self explanatory as it does not follow the laws of vector algebra. But then he said that a very very small angle (limit) can be considered as a vector because it has negligible...
  24. T

    Safety of small angle approximations

    Today I was doing a problem (physics) and ended up with the a differential equation dT = -µ T sin(dθ) (Where µ is constant) I wasn't sure what to do with the sin(dθ), so I used the small angle approximation of sin(dθ) = dθ. I would think this would be a perfect approximation because dθ is...
  25. J

    Differential Cross-Section for Small Angle Scattering in a Potential Field

    Homework Statement Find the differential cross-section for small-angle scattering in a field U(r)=a/sqrt(b^2 + r^2)Homework Equations (let p be the greek letter ro, s be sigma, t be theeta) the general formula for small angle scattering is: Equation 1: ds=absolute value(dp/dt)*(p(t)/t)*do...
  26. S

    Is there a rule of thumb for small angle approximation?

    When you are not given an acceptable level of error in a problem, is there any rule of thumb I should use for how large Theta can be before I stop using the small angle approximation(Sin Theta=Theta) ?
  27. F

    Small angle formula question

    Basically this formula relates the angular size of an object (how big the object appears to an observer), the actual physical size d of the object, and the distance D from the observer to the object. So let's say i want to find the physical size of Betelgeuse. As a scientist figuring this...
  28. G

    Solving S_N Integral with Small Angle Formula

    Homework Statement S_N (x) = \frac{2}{\pi} \int_0^x \frac{\sin (2 N t )}{\sin (t)} \; d{t} use suitable small angle formula to showS_N \Big( \frac{\pi}{2 N} \Big) = \frac{2}{\pi} \int_0^{\pi} \frac{\sin u}{u} d{u} Homework Equations i guess the suitable small angle formula is \sin...
  29. G

    Small angle forula problem

    Homework Statement S_N(x)= \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\sin ((2 n-1)x)}{2 n-1} By considering a suitable small angle formula show that the value of the sum at this point is S_N \Big( \frac{\pi}{2 N} \Big)=\frac{2}{\pi} \int_0^{\pi} \frac{\sin (\mu)}{\mu} \; d{\mu}...
  30. K

    Single-slit diffraction and small angle approximation

    Homework Statement Light of wavelength 587.5 nm illuminates a single 0.75 mm wide slit. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum...
  31. quasar987

    Small angle approximation.

    In the context of a Lagrangian mechanics problem (a rigid pendulum of length l attached to a mass sliding w/o friction on the x axis), I found the following equations of motion and now I must solve them in the small oscillation limit. (I know the equations are correct)...
  32. A

    Small Angle Scattering: Electrons & Angles

    Why are electrons elastically scattered at very small angles but not large angles?
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